Tetrahedral cupola
# Tetrahedral cupola

Tetrahedral cupola :

| Schlegel diagram |||
| - | - | - |
| Type | Polyhedral cupola | Polyhedral cupola |
| Schläfli symbol | {3,3} v rr{3,3} | {3,3} v rr{3,3} |
| Cells | 16 | 1 rr{3,3} 1+4 {3,3} 4+6×{3} |
| Faces | 42 | 24 triangles 18 squares |
| Edges | 42 | 42 |
| Vertices | 16 | 16 |
| Dual | | |
| Symmetry group | [3,3,1], order 24 | [3,3,1], order 24 |
| Properties | convex, regular-faced | convex, regular-faced |

In 4-dimensional geometry, the tetrahedral cupola is a polychoron bounded by one tetrahedron, a parallel cuboctahedron, connected by 10 triangular prisms, and 4 triangular pyramids.

## Related polytopes

The tetrahedral cupola can be sliced off from a runcinated 5-cell, on a hyperplane parallel to a tetrahedral cell. The cuboctahedron base passes through the center of the runcinated 5-cell, so the Tetrahedral cupola contains half of the tetrahedron and triangular prism cells of the runcinated 5-cell. The cupola can be seen in $A_2$ and $A_3$ Coxeter plane orthogonal projection of the runcinated 5-cell:

$A_3$ Coxeter plane :

| Runcinated 5-cell | Tetrahedron (Cupola top) | Cuboctahedron (Cupola base) |
| - | - | - |
| $A_2$ Coxeter plane |||
